Philosophy Lexicon of Arguments

 
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I 52
Russell's antinomy/Waismann: The quantity of all humans is not a human, but the set of all concepts is a concept. It therefore contains itself, not normally.
The quantity of all humans does not contain itself, normal. "N". Let us ask whether "N" is normal or not; i.e. whether it contains itself or not!
Suppose, initially, N contains itself as an element, then the set N occurs among its elements. Thus, N contains a non-normal set, which is N, whereas, by definition, it should contain only normal sets. The assumption was therefore wrong.
Thus only the opposite can be true, but this also leads to a contradiction: If N does not contain itself as an element, then N is a normal set.
However, since N should contain all normal sets, it must also contain the normal set N, i.e. containing itself - but this is again a contradiction. It follows from the concept of the set itself.


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Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976


> Counter arguments against Waismann

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Ed. Martin Schulz, access date 2017-09-23